This invention is in the field of data communications, and is more specifically directed to modulator/demodulators (modems) for use in such communications.
As is well-known in the art, many modern data communications utilize multicarrier modulation (MCM) to provide high data rates. According to this technology, multiple carrier-modulated data streams are transformed into a single waveform for transmission; the inverse transformation at the receiver separate the multiple carriers, and recover the modulating symbols. MCM techniques are used in such communications applications as such as Digital Subscriber Line (DSL) services, fixed wireless, digital audio broadcast (DAB), and terrestrial digital video broadcast (DVB-T). MCM fundamentals are described in Bingham, “Multicarrier Modulation for Data Transmission: An Idea Whose Time Has Come”, IEEE Communications Magazine (May, 1990), pp. 5–14, and Cioffi, A Multicarrier Primer, T1E1.4/91–157, (Amati Comm. Corp. and Stanford University, November 1991).
One important and now popular modulation standard for DSL communication is Discrete Multitone (DMT). According to DMT technology, the available spectrum is subdivided into many subchannels (e.g., 256 subchannels of 4.3125 kHz). Each subchannel is centered about a carrier frequency that is phase and amplitude modulated, typically by Quadrature Amplitude Modulation (QAM), in which each symbol value is represented by a point in the complex plane; the number of available symbol values depends, of course, on the number of bits in each symbol. During initialization of a DMT communications session, the number of bits per symbol for each subchannel (i.e., the “bit loading”) is determined according to the noise currently present in the transmission channel at each subchannel frequency and according to the transmit signal attenuation at that frequency. For example, relatively noise-free subchannels may communicate data in ten-bit to fifteen-bit symbols corresponding to a relatively dense QAM constellation (with short distances between points in the constellation), while noisy channels may be limited to only two or three bits per symbol (to allow a greater distance between adjacent points in the QAM constellation). In this way, DMT maximizes the data rate for each subchannel for a given noise condition, permitting high speed access to be carried out even over relatively noisy twisted-pair lines.
DMT modulation also permits much of the processing of the data to be carried out in the digital domain. Typically, the incoming bitstream is serially received and then arranged into symbols, one for each subchannel (depending on the bit loading). Reed-Solomon coding and other coding techniques are also typically applied for error detection and correction. Modulation of the subchannel carriers is obtained by application of an inverse Discrete Fourier Transform (IDFT) to the encoded symbols, producing the output modulated time domain signal. This modulated signal is then serially transmitted. All of these operations in DMT modulation can be carried out in the digital domain, permitting implementation of much of a DSL modem, and particularly much of the processing-intensive operations, in a single chip (such as a Digital Signal Processor, or DSP).
The discrete output time domain signal from the modulation is then converted into a time-domain analog signal by a conventional digital-to-analog converter. The analog signal is then communicated over the transmission channel to the receiving modem, which reverses the process to recover the transmitted data. The non-ideal impulse response of the transmission channel of course distorts the transmitted signal. Accordingly, the signal received by the receiving modem will be a convolution of the analog output waveform with the impulse response of the transmission channel. Ideally, the DMT subchannels in the received signal are orthogonal so that the modulating data can be retrieved from the transmitted signal by a Discrete Fourier Transform (DFT) demodulation, under the assumption that convolution in the time domain corresponds to multiplication in the frequency domain.
One may express the time-domain signal y(t) at the receiver, based on a transmitted time-domain signal x(t), as:y(t)=x(t){circle around (x)}h(t)This expression simply states that the received signal y(t) is the time-domain convolution of the input signal x(t) with the channel impulse response h(t). In the ideal case, this time-domain expression can be expressed in the frequency-domain as:Y(ω)=X(ω)·H(ω)where X(ω), H(ω), and Y(ω) are the respective frequency-domain representations of time-domain signals x(t), h(t), y(t). Considering that the transmitted signal x(t) is the IDFT of the symbol sequences at their respective subchannel frequencies, the frequency-domain spectrum X(ω) corresponds to the symbols themselves. According to the DMT modulation technology, the receiver can therefore retrieve the symbols X(ω) by removing the channel response H(ω) from the DFT of the frequency-domain received signal Y(ω); this can typically be performed by a single-tap frequency domain equalizer.
However, time domain convolution corresponds to frequency domain multiplication only if the input sequence is infinitely long, or if the input sequence is periodic. Because the number of subchannels is finite, however, the number of real-valued time-domain samples at the output of the transmitter IDFT (i.e., the “block” length) is also finite. Accordingly, it is useful to make the transmitted signal appear to be periodic, at a period on the order of the block length. A well-known technique is the use of a cyclic prefix in the transmitted data stream. The cyclic prefix is generally defined as a number ν of samples at the end of a block of samples in the output bitstream. These ν samples are prepended to the block, prior to digital-to-analog conversion. This effective periodicity in the input sequence thus permits the use of a DFT to recover the modulating symbols in each subchannel, under the assumption that the cyclic prefix length is less than the channel length.
In effect, the use of the cyclic prefix eliminates inter-symbol interference (ISI) between adjacent data frames, and inter-carrier interference (ICI) between subchannels. ISI generally arises from distortion and spreading of the transmitted signal over the channel, which causes the end of one DMT symbol to overlap into the beginning of the next DMT symbol. ICI affects the independence of the subcarriers, resulting in loss of orthogonality among the subchannels. Of course, if the subchannels are no longer orthogonal to one another, the modulating data on these subchannels cannot be separated at the receiver.
However, in order for the input sequence to truly appear periodic, and for the ISI interference to be contained within the redundant prefix of the block, the cyclic prefix must be longer than the length of the channel response. To ensure that ISI is not present in the transmitted signal, therefore, one may of course use a long cyclic prefix. Because the cyclic prefix does not itself contain any information or “payload” (considering that the prefix is redundant with the samples at the end of the block), a cyclic prefix of any length reduces the efficiency of the transmission. Accordingly, some transmission channels may have impulse responses that are so long as to prevent DMT data transmission at a reasonable efficiency.
In many DSL subscriber loops, the transmission channel response may indeed be very long, requiring an extremely long cyclic prefix, and resulting in a significant loss of data rate. By way of further background, time domain equalizers are known in the art as useful in effectively reducing the length of the channel response, for example as described in Chow et al., “A Discrete Multitone Transceiver System for HDSL Applications”, IEEE Journal on Selected Areas in Communications, Vol. 9, No. 6 (August 1991), pp. 895–908. A time domain equalizer is typically a small finite impulse response digital filter at the receiver that is applied to the received signal after conversion from analog to digital, but prior to removal of the cyclic prefix and prior to the DFT into the frequency domain (hence the name time domain equalizer). The time domain equalizer is intended to reduce the effective length of the channel response to less than the length of the cyclic prefix.
FIG. 1 illustrates an algorithm for defining the response of a time domain equalizer according to the conventional minimum mean squared error (MMSE) technique. In the MMSE algorithm, an input signal x(t) is applied to channel response h(t) model 2, which may be derived from an estimate of the channel response. The application of the input signal x(t) to channel response model 2 is then added to a noise estimate n(t), via adder 3, to produce received output signal y(t), which is applied to time domain equalizer 4. Time domain equalizer 4 may be implemented as an adaptive digital filter, as will be described below. The output of time domain equalizer 4 is applied to one input of adder 5. The input signal x(t) is also applied, through delay stage 6 (which compensates for the delay through the other leg), to desired impulse response model 8. The desired impulse response of model 8 is selected to be of a length that is less than the length of the cyclic prefix to be used. The output of desired impulse response model 8 is applied to a negative input of adder 5, which generates an error signal e(t) at its output that is applied to time domain equalizer 4.
In defining the coefficients to be used in a digital filter representation of time domain equalizer 4, time domain equalizer 4 is iteratively adjusted in response to the error signal e(t), in a direction to minimize error signal e(t). Upon convergence, the output of time domain equalizer 4 corresponds to the input signal x(t) convolved with the desired impulse response of model 8, from which the input signal x(t) can be readily recovered. The coefficients of the digital filter used to realize time domain equalizer 4 at this converged state can then be applied as a time domain equalizer in a receiving modem, with the equalizer then serving to reduce the effective length of the transmission channel response to the desired length, preferably within the length of the cyclic prefix. The combination of the channel response h(t) and the time domain equalizer filter is often referred to as the target impulse response, or TIR.
The convergence algorithm can be solved by a convenient matrix formulation. A vector w=[w0w1 . . . wl]T can be defined as the l+1 sample TEQ impulse response, and a vector b=[b0b1 . . . bk]T can be defined as the k+1 sample target channel. Where y=[y0y1 . . . y−1]T is the last l+1 samples of the channel output, and xΔ=[x−Δx−Δ−1 . . . x−Δ−k]T is defined as a delayed input sequence, the error e[n] for n=0 can be derived as:e[0]=wTy−bTxΔwhich corresponds to the difference between the outputs of the two channels form TEQ 4 and target channel response 8 of FIG. 1, as produced by adder 5. The mean-squared error E[e2[0]] can thus be given by:E[e2[0]]=wTRyyw+bTRxxb−2wTRyxwwhere Rxx and Ryy are the autocorrelation matrices for xΔ and y, respectively, and where Ryx is their cross-correlation matrix.
In practice, minimization of the mean-squared error E[e2[n]] computes the values of filter matrices w and b, and time delay Δ, that minimizes the function. In order to avoid the trivial solution of zero-valued matrices w and b, either a unit tap constraint (UTC) on matrix b (bk=1) or a unit norm constraint (UNC) on matrix w (∥w∥=1) is applied. Standard Lagrange multiplier techniques derive a solution in either case, depending upon the matrix RMSE=Ryy−RyxRxx−1Rxy. For the UNC case, the time domain equalizer vector w becomes the eigenvector of RMSE that is associated with the minimum magnitude eigenvalue. In the UTC case, the time domain equalizer is:
            ω      _        UTC    =                    R        MSE                  -          1                    ⁢                        δ          _                k                                      δ          _                k        T            ⁢              R        MSE            ⁢                        δ          _                k            where δk=1 for sample k, and zero elsewhere.
Variations on the MMSE implementation of a time domain equalizer are known in the art. Van Kerckhove et al., “Adapted Optimization Criteria for FDM-based DMT-ADSL Equalization”, ICC 1996, pp. 1328–34 describes an approach in which the MMSE minimization of the time domain equalizer is tuned for an FDM-based ADSL modem and is performed with the injection of virtual noise; the noise signal is virtual in that it is mathematically generated rather than based on noise measurements.
In most asymmetric DSL (ADSL) applications, as is well known, upstream (subscriber to central office) communications are in a low frequency band while downstream (central office to subscriber) communications are in a high frequency band, where the two frequency bands do not overlap one another; this approach is referred to as frequency division multiplexing (FDM). Typically, therefore, analog filtering is performed at the receiver to cancel echoes of its upstream signal from interfering with the received downstream signal. This analog filtering is a significant contributor to the overall target impulse response (TIR), especially if the high-pass downstream filter is required to have a sharp rolloff characteristic.
Conventional optimization of a time domain equalizer in an ADSL modem therefore involves difficult tradeoffs. In theory, a time domain equalizer can be derived that compensates both for a long channel response and also a sharp high-pass filter characteristic, however such a conventional time domain equalizer will necessarily be quite complex. In addition, the resulting TEQ will add significant energy into the low-frequency band due to the high-pass filtering, which will necessarily amplify noise in this band and cause it to spread into the low-frequency symbols, reducing the signal-to-noise ratio and also impacting resolution of the TEQ coefficients for high frequency components.
By way of further background, it has been observed that the frequency domain behavior of conventional MMSE TEQs when applied to FDM modems is quite poor. Near nulls often result in the passband region, while the downstream DFT demodulation process causes appreciable spectral leakage of noise power between adjacent frequency bins, including into the near null frequencies. These near nulls therefore result in a loss of channel capacity. To address this issue, alternative MMSE design of the time domain equalizer is based on the maximizing of an approximation to the system channel capacity. According to this approach, the time domain equalizer is designed by maximizing a product of the power spectra of the target channel over the DFT bins in the passband regions of the received signal, as described in Al-Dhahir et al., “Optimum Finite Length Equalization for Multicarrier Transceivers”, IEEE Trans. Comm., Vol. 44, No. 1 (January 1996), pp. 56–64. In effect, this maximization eliminates near nulls in the spectrum of the TEQ within the pass band. Because the design of time domain equalizers using this criterion requires a computationally complex non-linear constrained optimization procedure, approximations to this approach have been developed. According to Farhang-Boroujeny et al., “An Eigen-Approach to the Design of Near-Optimum Time Domain Equalizers for DMT Transceivers”, ICC 1999, an ad hoc approximation to the capacity maximization problem is to define the time domain equalizer from a linear combination of the eigenvectors of the correlation matrix of the TEQ input that provides a TIR with no null in its spectrum and provides a relatively low MSE.